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The ARIANNA detector is designed to detect neutrinos with energies above 10(17) eV. Due to the similarities in generated radio signals, cosmic rays are often used as test beams for neutrino detectors. Some ARIANNA detector stations are equipped with antennas capable of detecting air showers. Since the radio emission properties of air showers are well understood, and the polarization of the radio signal can be predicted from the arrival direction, cosmic rays can be used as a proxy to assess the reconstruction capabilities of the ARIANNA neutrino detector. We report on dedicated efforts of reconstructing the polarization of cosmic-ray radio pulses. After correcting for difference in hardware, the two stations used in this study showed similar performance in terms of event rate and agreed with simulation. Subselecting high quality cosmic rays, the polarizations of these cosmic rays were reconstructed with a resolution of 2.5 degrees (68% containment), which agrees with the expected value obtained from simulation. A large fraction of this resolution originates from uncertainties in the predicted polarization because of the contribution of the subdominant Askaryan effect in addition to the dominant geomagnetic emission. Subselecting events with a zenith angle greater than 70 degrees removes most influence of the Askaryan emission, and, with limited statistics, we found the polarization uncertainty is reduced to 1.3 degrees (68% containment).